On dB spaces with nondensely defined multiplication operator and the existence of zero-free functions

TL;DR

This paper investigates de Branges spaces with non-densely defined multiplication operators, characterizing their selfadjoint extensions and establishing conditions for zero-free functions to belong to these spaces.

Contribution

It introduces a novel analysis of selfadjoint extensions of the multiplication operator in de Branges spaces and provides new criteria for zero-free functions to be included.

Findings

Characterization of selfadjoint extensions as rank-one perturbations

New necessary and sufficient conditions for zero-free functions in de Branges spaces

Insights into the structure of de Branges spaces with non-densely defined operators

Abstract

In this work we consider de Branges spaces where the multiplication operator by the independent variable is not densely defined. First, we study the canonical selfadjoint extensions of the multiplication operator as a family of rank-one perturbations from the viewpoint of the theory of de Branges spaces. Then, on the basis of the obtained results, we provide new necessary and sufficient conditions for a real, zero-free function to lie in a de Branges space.